c. Proof that, if a function \( f(x) \) has a power series representation centered at \( x=a \) that converges to \( f(x) \) on some open interval containing \( a \), then this power series a Taylor series for \( f(x) \) at \( x=a \).

To prove that if a function \( f(x) \) has a power series representation centered at \( x=a \) that converges to \( f(x) \) on some open interval containing \( a \), then this power series is a Taylor series for \( f(x) \) at \( x=a \), we need to show that the power series representation of \( f(x) \) centered at \( x=a \) is the same as the Taylor series representation of \( f(x) \) at \( x=a \). The Taylor series representation of a function \( f(x) \) at a point \( x=a \) is given by: \[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \] where \( f'(a) \), \( f''(a) \), \( f'''(a) \), and so on are the derivatives of \( f(x) \) evaluated at \( x=a \). Given that \( f(x) \) has a power series representation centered at \( x=a \) that converges to \( f(x) \) on some open interval containing \( a \), we can write the power series representation of \( f(x) \) as: \[ f(x) = \sum_{n=0}^{\infty} a_n (x-a)^n \] where \( a_n \) are the coefficients of the power series. To prove that this power series is a Taylor series for \( f(x) \) at \( x=a \), we need to show that the coefficients \( a_n \) are the same as the coefficients in the Taylor series representation of \( f(x) \) at \( x=a \). The coefficients in the Taylor series representation of \( f(x) \) at \( x=a \) are given by: \[ a_n = \frac{f^{(n)}(a)}{n!} \] where \( f^{(n)}(a) \) is the \( n \)th derivative of \( f(x) \) evaluated at \( x=a \). Therefore, to prove that the power series representation of \( f(x) \) centered at \( x=a \) is a Taylor series for \( f(x) \) at \( x=a \), we need to show that the coefficients \( a_n \) in the power series representation are the same as the coefficients in the Taylor series representation. This completes the proof that if a function \( f(x) \) has a power series representation centered at \( x=a \) that converges to \( f(x) \) on some open interval containing \( a \), then this power series is a Taylor series for \( f(x) \) at \( x=a \).